Multiwavelet troubled-cell indicator for discontinuity detection of discontinuous Galerkin schemes

摘要

In this paper, we introduce a new global troubled-cell indicator for thediscontinuous Galerkin (DG) method in one- and two-dimensions. This is done bytaking advantage of the global expression of the DG method and re-expanding itin terms of a multiwavelet basis, which is a sum of the global average andfiner details on different levels. Examining the higher level differencecoefficients acts as a troubled-cell indicator, thus avoiding unnecessaryincreased computational cost of a new expansion. In two-dimensions themultiwavelet decomposition uses combinations of scaling functions andmultiwavelets in the x- and y-directions for improved troubled-cell indication.By using such a troubled-cell indicator, we are able to reduce thecomputational cost by avoiding limiting in smooth regions. We present numericalexamples in one- and two-dimensions and compare our troubled-cell indicator tothe subcell resolution technique of Harten (1989) and the shock detector ofKrivodonova, Xin, Remacle, Chevaugeon, and Flaherty (2004), which werepreviously investigated by Qiu and Shu (2005).